c
2019 by Institut Mittag-Leffler. All rights reserved
Strong property (T ) for higher-rank lattices
by
Mikael de la Salle
Centre National de la Recherche Scientifique Ecole Normale Sup´´ erieure de Lyon
Lyon, France
1. Introduction
Kazhdan’s property (T) is a rigidity property for unitary representations of a locally com- pact group, which has found numerous applications in various areas of pure and applied mathematics, see [3]. Vincent Lafforgue’s strong property (T) is a strengthening of prop- erty (T) which deals with representations by bounded operators with small exponential growth of the norm. Its introduction in [13] was motivated by the Baum–Connes conjec- ture, as it is a natural obstruction to apply Lafforgue’s approach to the Baum–Connes conjecture, see [15]. It has also found several applications, notably its Banach-space version that we will discuss below, as it provided the first examples of super-expanders (expanders which do not coarsely embed into any uniformly convex Banach space), and as it implies strong fixed point properties for affine actions on Banach spaces. Another notable recent application is also to dynamics, as it was one of the steps in spectacular progresses on the Zimmer program [7].
So far, strong property (T) has been shown for higher-rank connected simple Lie groups (or higher-rank simple algebraic groups over local fields) and their cocompact lattices. The case when the Lie algebra containssl3was proven by Lafforgue in [13]. The generalization to other algebraic groups was done by Liao [16] (for non-Archimedean local fields) and de Laat and the author [12] (for Archimedean local fields, i.e.R). In particular before the present work it was not known whether SL3(Z) has strong property (T). The aim of this article is to extend these results to cover the lattices which are not cocompact (for example SL3(Z)) as well. This will have consequences on the Zimmer program [8].
We also take the opportunity to state and prove all the results more generally for (lattices
Mikael de la Salle was supported by ANR grants GAMME and AGIRA.
in) semisimple groups rather than simple groups, and also to some non semisimple Lie groups (Remark4.3).
In the whole article, local field will mean commutative, non-discrete locally com- pact topological field. So a local field is a finite extension of R (in which case it is Archimedean), or of Qp or Fp((t)) for some prime number p(in which case it is non- Archimedean). Higher-rank simple group will mean either real connected simple Lie group of real rank >2, or connected almost F-simple algebraic group of F-split rank
>2 over a local field F. Higher-rank group will stand for a finite product of higher- rank simple groups. We warn the reader that for us, products of rank-1 groups such as SL2(R)×SL2(Qp) are not of higher rank. We refer to [18, Chapter I] for the terminol- ogy. Note that real connected simple Lie group or real rank >2 is more general than connected almost simple algebraic group of split rank>2 overR. It includes for example some groups with infinite center, as the infinite covering group of Sp2n(R).
Recall that a lattice in a locally compact groupGis a discrete subgroup Γ such that G/Γ carries aG-invariant Borel probability measure.
Theorem 1.1. Every lattice in a higher-rank group has strong property (T).
Examples of lattices in higher-rank groups include SLn(Z), SLn(Fp[t]), SLn(Z[1/p]), for n>3, and Sp2n(Z), fSp2n(Z) (the preimage of Sp2n(Z) in the universal cover of Sp2n(R)), Sp2n(Fp[X]), for n>2. None of these examples is a cocompact lattice, so for all these cases Theorem1.1is new.
When Γ is a cocompact lattice in a locally compact groupG, every representation of Γ by bounded operators on a Hilbert (or Banach) space can be induced in a satisfactory way to a representation ofG by bounded operators on a Hilbert (Banach) space. This is what allows one to prove that (Banach) strong property (T) passes to cocompact lattices; see [13]. As we shall explain in §2.2, when Γ is not cocompact, induction of representations which are not uniformly bounded does not behave well, and we do not see any reasonable way to define an induced Banach space representation. So the proof of Theorem1.1 does not proceed by proving that strong property (T) passes to lattices.
And we still have no idea whether such a statement is true (although amusingly, it is true that the negation of strong property (T) passes to lattices; see Corollary5.9). This might appear at first sight a bit surprising, because it is now very well understood (this seems to go back at least to the proof of the normal subgroup theorem by Margulis) that, although they might not be cocompact, higher-rank lattices are very much integrable (for example they areLp-integrable for every p<∞in the sense of [23]), and these good integrability properties enable to induce in a satisfactory way cocycles with values in isometric representations. The new idea that we introduce to overcome this difficulty is
a form of induction of representation π: Γ!GL(X) which, under some assumption on the integrability of the lattice and the growth rate of the norm ofkπ(γ)k, produces a representation-like object, where one is only allowed to compose once, and that we call a two-step representation.
Definition 1.2. Atwo-step representation of a topological groupGis a tuple (X0, X1, X2, π0, π1),
whereX0,X1 andX2 are Banach spaces andπi:G!B(Xi, Xi+1) are strongly continu- ous(1)maps such that
π1(gg0)π0(g00) =π1(g)π0(g0g00) for every g, g0, g00∈G.
In this case, we will denote byπ:G!B(X0, X2) the continuous map satisfying π(gg0) =π1(g)π0(g0) for every g, g0∈G.
It turns out that a form of strong property (T) also holds for two-step representations of higher-rank groups. And this property passes to undistorted lattices (Theorem 5.4).
This is the content of our main result Theorem 1.3, which contains Theorem 1.1 as a particular case.
Before stating it, we recall the notion of length function that we use, which contains as its main examples the word-length with respect to compact symmetric generating sets.
Alength function on a locally compact topological groupGis a function`:G!R+such that
• ` is bounded on compact subsets ofG;
• `(g−1)=`(g) for everyg∈G;
• `(gh)6`(g)+`(h) for everyg, h∈G.
The exponential growth rate of a two-step representation (X0, X1, X2, π0, π1) with respect to a length function`is
i=0,1maxlim sup
`(g)!∞
logkπi(g)k
`(g) .
We say that a pair (G, `) of a locally compact group with a length function satisfies (∗) if there exists s, t, C >0 and a sequence mn of positive probability measures whose support is contained in{g:`(g)6n}such that the following holds. Let (X0, X1, X2, π0, π1)
(1) I.e., for everyx∈Xi, the mapG3g7!π(g)x∈Xi+1is continuous; see§2.
be a two-step representation andL a real number such that X1 is a Hilbert space and kπi(g)k6Les`(g)for allg∈Gandi∈{0,1}. Then, there isP∈B(X0, X2) such that
kπ(mn)−Pk6CL2e−tn, (1.1)
and such that
limn kπ(δg∗mn∗δg0)−π(mn)k= 0 for every g, g0∈G. (1.2) Strong property (T) corresponds to the case when X0=X1=X2=X and π is a repre- sentation. In that case, (1.2) is usually replaced by the equivalent property that P is a projection on the space of invariant vectors{x∈X:π(g)x=xfor allg∈G}, parallel to a π(G)-invariant complement subspace. The condition (1.2) is nothing but a reformu- lation which remains meaningful in the above generality when there is no such thing as invariant vector or projection.
We say that G satisfies (∗) if (G, `) satisfies (∗) for every length function `, or equivalently if G is compactly generated and (G, `) satisfies (∗) for the word-length function coming from a compact generating set. See Lemma2.6for the equivalence.
Theorem 1.3. Every higher-rank group or lattice in it satisfies property(∗).
Examples of mapsπas in (∗) are whenX is a topological vector space (for example the space of measurable functions on a manifold, or just a measure space) and π:G! GL(X) is a continuous representation of Gon X which a priori does not preserve any Banach space inX (for example because of losing of derivatives, as in the Nash–Moser theorem, or of integrability). But there are three Banach spacesX0, X1, X2 with X1 a Hilbert space with continuous embeddings intoX (for example encoding different scales of derivability or integrability) and such thatπ(g) mapsXitoXi+1with norm6Les`(g). In that situation, we can apply the conclusion of the theorem. In particular, we get, for every x∈X0, thatπ(mn)x converges in the norm of X1 (and hence in the topology of X) to aπ(G)-invariant vector. In this setting, property (∗) has therefore to be seen as a procedure to systematically produce and locate invariant vectors inX.
I would like to point out that, even if one is only interested in strong property (T) (so to representations on Hilbert spaces), it is crucial that in property (∗) we allow arbitrary Banach spacesX0 and X2. Indeed, the induction procedure explained in§5.2, which is the heart of this work, cannot produce Hilbert spaces but more general Banach spaces (namely Hilbert-space valuedLp spaces for various values ofp).
Banach space extensions
Higher-rank groups over non-Archimedean local fields and their cocompact lattices are known to satisfy strong Banach property (T) with respect to every class of Banach spaces of non-trivial (Rademacher) type [14], [16] (see §2 for the definitions). Moreover, this class is essentially the optimal class. Although some partial results have been obtained [21], [12], [11], it is still not known whether the same holds over the real numbers. I regard this question as the main open problem on the subject, as a positive answer would settle positively the conjecture in [1] that every action by isometries on a uniformly convex Banach space of a higher-rank lattice has a fixed point, and prove that the standard Cayley graphs of SL3(Z/nZ) form a family of super-expanders.
In this article we also extend to all lattices the above mentioned results.
To state the results, we introduce the following notion: if E is a class of Banach spaces we say thatG(respectively (G, `)) satisfies (∗E) if in (∗) the assumption thatX1
is a Hilbert space is replaced byX1∈E.
The following result extends the results of Lafforgue and Liao [14], [16].
Theorem 1.4. Let Gbe a higher-rank simple group over a non-Archimedean local field, or a lattice therein. Then, G satisfies (∗E) for every class of Banach space E of non-trivial type.
In particular, every lattice in a higher-rank group over non-Archimedean local fields has strong property (T) with respect to every Banach space of non-trivial type.
In the real case, the conditions we have to impose on the Banach spaces are a bit longer to state, but we believe that they are equivalent to having non-trivial type. For n>2, denote bySn the unit sphere in EuclideanRn+1 and define a family (Tδ(n))δ∈[−1,1]
of operators onL2(Sn) byTδ(n)f(x) is the average off on{y∈Sn:hx, yi=δ}.
Forθ∈R/2πZ, denote bySθ the operator onL2(SU(2)) given by Sθf(u) =
Z 2π 0
f 1
√2
e−iθ −eiϕ e−iϕ eiθ
u
dϕ 2π.
The following result extends the results of [21], [12], [11]. A version for general higher-rank groups is stated as Theorem5.11.
Theorem 1.5. Let Gbe a connected simple Lie group with Lie algebra gand Γ⊂G be a lattice. Then both Gand Γ have (∗E) (and therefore strong (T) with respect to E) if one of the following conditions holds:
• g contains a Lie subalgebra isomorphic to sp4, and there is α∈(0,1] and C >0 such that,for every X∈E,
kSθ−Sπ/4kB(L2(SU(2);X))6C θ−14π
α/4 for all θ∈[0,2π] (1.3)
and
kTδ(2)−T0(2)kB(L2(Sn;X))6C|δ|α/2 for all δ∈[−1,1]. (1.4)
• gcontains a Lie subalgebra isomorphic to sl3n−3 for n>2,and there is α∈(0,1]
and C >0 such that, for every X∈E,
kTδ(n)−T0(n)kB(L2(Sn;X))6C|δ|α/2 for all δ∈[−1,1]. (1.5)
All the conditions (1.3)–(1.5) imply that X has non-trivial type, and we believe that they are actually all equivalent. However, we only know that the condition when g contains sp4 is formally stronger than when it contains sl3, and the condition (1.5) becomes formally weaker whenngrows. WhenX is a Hilbert space, both (1.3) and (1.4) hold withα=1. Therefore, (1.3) and (1.4) hold ifX is isomorphic to a subspace of an interpolation space [X0, X1]αbetween a Hilbert spaceX1and an arbitrary Banach space X, or more generally if X is θ-Hilbertian (with θ=α) in the sense of [20]. This holds in particular ifX is isomorphic a subspace of a super-reflexive Banach lattice [19]. This includes for example all reflexive Sobolev spaces or Besov spaces.
Since every real simple Lie algebra of real rank >2 contains a Lie subalgebra iso- morphic to sl3 or sp4, the preceding implies that every higher-rank lattice has strong (T) with respect toθ-Hilbertian Banach spaces, but the results are more general as they include some non super-reflexive spaces, for example those having good enough type and cotype exponents; see [21].
We end this introduction with another particular case of the above theorem (see [11]
for the proof that the assumption in Corollary 1.6 implies that (1.5) holds for n large enough).
Corollary1.6. Let X be a Banach space for which there isβ <12 andCsuch that, for every integer k, every subspace of X of dimension k is at Banach–Mazur distance 6Ckβ from `k2. There is NX such that every lattice in a connected simple Lie group of real rank >NX has strong property (T)with respect to X.
Theorem1.3, as well as its Banach space generalizations, is proven in several steps.
The first step is to prove the theorem for the basic building blocks of higher-rank groups, namely forG=SL3(F), Sp4(F) forF=R,Qp orFp((t)), orG=fSp4(R). This is achieved in§3. The second step is to extend this to all higher-rank groups in§4. The last step is to deal with lattices in such groups in§5. A crucial ingredient is the fact that higher-rank lattices areexponentially integrable.
Acknowledgements
I thank David Fisher and Tim de Laat for many interesting conversations and useful com- ments. I thank Fran¸cois Maucourant for allowing me to include his proof of Theorem5.3, much more elementary and general than my initial argument based on the reduction the- ory of S-arithmetic lattices. Even though this argument is no longer present in the final version of this work, I wish to thank Olivier Ta¨ıbi and Kevin Wortman for their very patient explanations on the reduction theory of S-arithmetic lattices in positive characteristic.
2. Preliminaries 2.1. Notation
IfGis a locally compact group, we will denote byPc(G) the set of all compactly supported Borel probability measures onG. To lighten the notation, the convolution of probability measuresm1, m2∈Pc(G) will be written asm1m2. So,
Z
f d(m1m2) = Z Z
f(g1g2)dm1(g1)dm2(g2).
We viewPc(G) as a set of linear forms on the space of continuous functions on G, and equip it with the restriction of the weak-∗ topology.
If X and X0 are Banach spaces, a mapπ:G!B(X, X0) is called strongly continu- ous if, for everyx∈X, the mapG3g7!π(g)x∈X0 is continuous. In that case, for every m∈Pc(G), we denote byπ(m)∈B(X, X0) the operatorx7!R
π(g)x dm(g) (Bochner inte- gral). By applying the definitions, we readily obtain the following.
Lemma 2.1. If π:G!B(X, X0)is strongly continuous,then the map π:Pc(G)−!B(X, X0)
is still strongly continuous.
We recall the definition of Lafforgue’s strong property (T).
Fix a left Haar measuredgonG. If`is a length function on locally compact group G, denote by C`(G) the Banach algebra obtained by completion of convolution algebra Cc(G) under the norm kfk`=sup{kπ(f)k}, where the supremum is over all strongly continuous representations πof G on a Hilbert space for which kπ(g)k6e`(g) for every g∈G. As for measures,π(f) is here the operator x7!R
f(g)π(g)x dg.
For example, if `=0, we obtainC∗(G), the fullC∗-algebra ofG.
Definition 2.2. (Lafforgue) A locally compact groupG has strong property (T) if, for every length function`, there existss>0 such that, for everyc>0, the Banach algebra Cs`+c(G) has aKazhdan projection, i.e. an idempotentP such thatπ(P) is a projection on the space of invariant vectors for every representation π satisfying kπ(g)k6es`(g)+c for everyg∈G.
A justification for this definition is the following well-known characterization of prop- erty (T), which in particular asserts that the particular case`=0,c=0 in Definition2.2 is equivalent to property (T).
Proposition 2.3. For a locally compact group G, the following are equivalent:
(1) G has property (T).
(2) There is a compactly supported probability measure µon Gsuch that,for every unitary representation πof Gon a Hilbert,kπ(µ)−Pπk612, where Pπ is the orthogonal projection on the space of invariant vectors and the norm is the operator norm on G.
(3) G has a symmetric compact generating set Q and there is a sequence µn of probability measures supported in Qn such that,for every unitary representation πof G on a Hilbert space,kπ(µn)−Pπk62−n.
(4) For every length function ` on G, there are constants C, s>0 and a sequence µn of probability measures supported in {g∈G:`(g)6n}such that,for every unitary rep- resentation π of G on a Hilbert,kπ(µn)−Pπk6Ce−sn.
(5) C∗(G)has a Kazhdan projection.
Remark 2.4. Actually this proposition holdsrepresentation-by-representation: given a unitary representationπof a locally compact groupG, the following are equivalent:
• πhas spectral gap in the sense that the orthogonal of the space of invariant vectors does not carry almost invariant vectors.
• There is a compactly supported probability measureµonGsuch that kπ(µ)−Pπk612.
• There is a symmetric compact subset Q⊂G, and a sequence of probability mea- suresµonQn such that
kπ(µn)−Pπk62−n.
• For every length function`onG, there are constantsC, s>0 and a sequence µn
of probability measures supported in{g∈G:`(g)6n} such that kπ(µn)−Pπk6Ce−sn.
If one defines correctly a Kazhdan projection for arbitrary Banach-algebra completions ofCc(G) (see [22]), these definitions are in turn equivalent to the existence of a Kazhdan projection for the completion ofCc(G) for the normkfk=kπ(f)k.
If E is a class of Banach spaces, one can denote similarly by C`,E(G) the Banach algebra obtained by completion of Cc(G) under the norm kfk`,E=sup{kπ(f)k}, where the supremum is over all strongly continuous representationsπ ofGon a Banach space inE for whichkπ(g)k6e`(g)for everyg∈G, and define Banach strong property (T) with respect toE as strong property (T) by replacingCs`+c(G) byCs`+c,E(G).
Recall that a Banach spaceX has non-trivial Rademacher type (or simply non-trivial type) if there existsp>1 and a real numberT such that
E
X
i
εixi
p1/p 6T
X
i
kxikp 1/p
(2.1) for every finite sequencexiinX, whereεiare independent identically distributed random variables uniformly distributed in{−1,1}. This is equivalent to the fact that `1 is not finitely representable inX: there isN >0 andc>1 such that every linear mapubetween
`1N and everyN-dimensional subspace ofX satisfieskuk ku−1k>c.
More generally, a class of Banach spaces E has non-trivial type if there existsp>1 andT <∞such that (2.1) holds for everyX∈E and every finite sequence (xi)i inX, or equivalently if`1is not finitely representable in E.
2.2. Why the naive attempt does not work
We now explain why the classical notion of induction of representations, that we first recall, is not well suited to induce strong (T) to non-cocompact lattices.
Let Γ be a lattice in a locally compact group G. Let Ω be a Borel fundamental domain for G/Γ: Ω is a subset of Ω, belonging to the Borel σ-algebra, and such that Ω×Γ3(ω, γ)7!ωγ∈Gis a bijection.
Let π be a representation of Γ on a Hilbert or Banach space X. Consider the topological vector spaceXe of (Bochner-measurable) functionsf:G!X satisfying
f(gγ) =π(γ)−1f(g),
moded out by functions that vanish outside of a negligeable set. Make G act on this space by left translation: ˜π(g)f(h)=f(g−1h).
It is natural to consider the Hilbert space of such functions satisfying moreover Z
Ω
kf(ω)k2Xdω 1/2
<∞.
This space is naturally identified withL2(Ω;X). Under this identification, if gω= (g·ω)α(g, ω)
is the unique decomposition ofgωin G=ΩΓ, then ˜π(g) reads as
(˜π(g)f)(ω) =π(α(g−1, ω)−1)f(g−1·ω) for allg∈Gandω∈Ω.
The problem that occurs is that ˜π(g) preserves L2(Ω;X) if and only if the function ω7!kπ(α(g−1, ω)−1k is essentially bounded on Ω.
Lemma 2.5. The norm of π(g)˜ on L2(Ω;X)is equal to the essential supremum of ω7!kπ(α(g−1, ω)−1)k.
Proof. Let Cg be the essential supremum of kπ(α(g−1, ω)−1)k. The inequality k(˜π(g)f)(ω)k6kπ(α(g−1, ω)−1)k kf(g−1·ω)k implies that
k(˜π(g)f)(x)k2L
2(Ω;X)6Cg2 Z
Ω
kf(g−1·ω)k2dω=Cg2kfk2L
2(Ω;X), becauseω7!g−1·ω preserves the measure on Ω.
For the other direction, forγ∈Γ, denote
A={ω∈Ω :α(g−1, ω)−1=γ}= Ω∩gΩγ−1.
If A has positive measure, then for every x∈X we can consider f=χAx. It has norm
|A|1/2kxk, and its image χgAπ(γ)x has norm |A|1/2kπ(γ)xk. Taking the supremum overxyields the inequalityk˜π(g)k>kπ(γ)k. Taking the supremum over allg such that Ω∩gΩγ−1has positive measure prove thatk˜π(g)kis larger than or equal to Cg.
So, in general, ˜π is not a representation by bounded operators unless Γ is cocom- pact orπ is a uniformly bounded representation. There does not seem to be any other reasonable pseudo-norm on Xe for which ˜π(g) is by bounded operators. There is always the pseudo norm kfk=∞for allf6=0, but this is clearly unreasonable. We do not give a precise meaning to “reasonable”, but it should at least remember the whole represen- tation, for example by giving finite norm, for everyx∈X, to the constant function equal toxon Ω.
We mention however the construction in [9] where a pseudo-norm is constructed onXe, which, under the assumption that the bounded cohomologyHb1(Γ;π) is non-zero, gives rise to a non-zero space for whichHb1(G;Xe) is also non-zero.
2.3. Comparing Theorems 1.1and 1.3
We recall thatC`,E(G) has a Kazhdan projection if and only if there is a sequencemn of signed(2)compactly supported measures on Gwith R
1dmn=1 andC >0 such that kmn−mn+1k`,E6Ce−n and such that limnkgmn−mnk`,E=0 for every g∈G. Moreover, mncan be taken to be of the form (m1)n(thenth convolution power ofm1). In particular, mnis supported in{g:`(g)6nR}ifm1is supported in{g:`(g)6R}. Also, ifEis stable by duality and subspaces, then the preceding implies that limnkmng−mnk`,E=0 for every g∈G. For details, we refer to [22], where these assertions were established.
Hence, in the particular case whenX0=X1=X2=Xandπis a representation onX, property (∗) for (G, `) says a bit more than thatCs`+c(G) has a Kazhdan projection for every c>0: first it says that mn can be taken independent from c, that C=O(e2c) and most importantly thatmn can be taken to be positive.
2.4. Basic properties
The first basic lemma implies that to prove Theorem1.3, it is enough to consider the word-length function with respect to some compact symmetric generating set (which exists because (∗) is formally stronger than property (T), which already implies compact generation [3]), or any other length function quasi-isometric to it. Indeed, if ` is any length function on a locally compact compactly generated groupG, andQis a compact symmetric generating set for G with associated length function `Q, then there is a>0 such that`6a`Q. Namely, the supremum of` onQ.
Lemma 2.6. Let ` and `0 be two length functions, and a, b>0 such that `06a`+b.
If (G, `)has (∗E)then so does (G, `0).
Proof. If (G, `) has (∗E) withs, t, C and mn, it is immediate that (G, `0) has (∗E) withs/a,t/a,C0 andmb(n−b)/ac, whereC0=Ce(2sb+ta+tb)/a.
In each section of the paper, the proof of (∗) or (∗E) is divided in two parts: one first finds a sequencemnsuch that, ifs>0 is small enough andπis as in (∗), thenπ(mn) converges as in (1.1). Then one proves that (1.2) also holds. This second part is always much harder than the first one. The next remark shows that it is not necessary to prove the norm convergence in (1.2).
Remark 2.7. In (∗), condition (1.2) can be strengthened (or weakened). Indeed, once one knows that (1.1) holds for everyπ as in (∗), then, for any µ1, µ2∈Pc(G), one can apply it to the newπ0 given byπ0(m)=π(µ1mµ2). Indeed, thisπ0 satisfies the same
(2) It is not known in general ifmncan be taken to be positive.
assumptions, but withL replaced byLe(s/2)(R1+R2) if the support ofµi is contained in {g:`(g)6Ri}. And so there is µ1Pµ2∈B(X0, X2) such that, for every n,
kπ(µ1mnµ2)−µ1Pµ2k6CL2es(R1+R2)−tn. (2.2) So, (1.2) is equivalent to each of the following properties:
• δgPδg0=P;
• kπ(gmng0)−π(mn)k62CL2es(`(g)+`(g0))−tn;
• for everyx∈X0, limnkπ(gmng0)x−π(mn)xk=0;
• for everyx∈X0, limn(π(gmng0)x−π(mn)x)=0 weakly.
Lemma2.8. If (G1, `1)and(G2, `2)have(∗) (resp. (∗E)),then so does (G1×G2, `), where `(g1, g2)=max(`1(g1), `2(g2)).
Proof. Fori=1,2, letsi,ti,Ciandm(i)n be as in (∗E) forGi. Definemn=m(1)n ⊗m(2)n . By definition, it is a probability measure supported in{g∈G1×G2:`(g)6n}.
Letπ:G1×G2!B(X0, X2) be as in (∗E) forC ands. We claim that the conclusion of (∗E) holds ifs>0 is small enough.
We can compute
kπ(mn)−π(mn+1)k6kπ(m(1)n ⊗m(2)n )−π(m(1)n ⊗m(2)n+1)k +kπ(m(1)n ⊗m(2)n+1)−π(m(1)n+1⊗m(2)n+1)k.
By (1.1) applied to the mapG23g27!π(m(1)n ⊗δg2), ifs6s2the first term is dominated by 2C2L2e2sn−t2n. Similarly, if s6s1, the second term is dominated by 2C1L2e2s(n+1)−t1n. So, ifs=min 13t1,13t2, s1, s2
, then
kπ(mn)−π(mn+1)k6(2C1e2s+2C2)L2e−sn. This implies thatπ(mn) is Cauchy and that (1.1) holds witht=sand
C=2C1e2s+2C2
1−e−s .
The validity of (1.2) follows with a similar proof, taking into account Remark2.7.
3. Proof of Theorem 1.3for SL3 and Sp4
The aim of this section is to prove Theorems 1.3–1.5 for SL3, Sp4 and Spf4(R), and Theorem1.5for SL3n−3. As we shall see, the proofs use the same two main ingredients as the proofs of strong property (T): one is harmonic analysis in the maximal compact
subgroups, and the other is a careful exploration process of the Weyl chambers using some elementary moves coming from the maximal compact subgroup. These ingredients are the same, but they are combined in a different way. We will give a complete and essentially self-contained proof for SL3 and be much more sketchy for the other groups.
This allows us to divide the length of the paper by a factor of at least 2, and we believe that the interested reader will be able to fill the details. The proof for SL3(F) is essentially independent from the local field, but for a better readability we have chosen to first focus on the real case, and then explain the small changes that one has to make to deal with non-Archimedean local fields.
3.1. Case of SL3(R)
We prove the theorem forG=SL3(R). We denote byK=SO(3)⊂Gthe maximal compact subgroup. By Lemma2.6, it is enough to prove the theorem for the length function
`(g) = max(logkgk,logkg−1k),
wherek · kis the norm induced from the naturalK-invariant Euclidean norm onR3: k(s1, s2, s3)k= (s21+s22+s23)1/2.
More precisely, we will prove that (SL3(R), `) has property (∗) with the parameters s <14, t=12−2s and C= 100
1−4s,
andmn being anyK-biinvariant probability measure on{g:n−16`(g)6n}.
Let π as in Theorem 1.3 with s<14. Denote by d the distance on the compactly supported Borel probability measures onGdefined by
d(m, m0) =kπ(m)−π(m0)kB(X0,X2).
The following lemma lists the properties of d. In (3) and in the rest of the proof, λ stands for the left regular representation ofK. It is the representation onL2(K) given by λ(k)f(·)=f(k−1·) for every k∈K and f∈L2(K). The crucial property is (3). It is an incarnation for the compact group K of more general phenomenon: uniformly bounded two-step representations of amenable groups are governed by the left regular representation. We do not elaborate on this, as all we need is (3).
Lemma3.1. The distance dhas the following properties: (1) (Convexity)For every m1, m2∈Pc(G),
d 12(m1+m2),12(m01+m02)
612(d(m1, m01)+d(m2, m02)).
(2) (Lower-semicontinuity)If Q⊂Gis compact and mi (resp. m0i)is a net of prob- ability measures supported in Qand converging weak-∗to m (resp. m0),then
d(m, m0)6lim inf
i d(mi, m0i).
(3) If µ and µ0 are probability measures on K and g1, g2∈G, then d(δg1µδg2, δg1µ0δg2)6L2es`(g1)+s`(g2)kλ(µ−µ0)kB(L2(K)).
Proof. Property (1) is obvious, and (2) is immediate from the strong continuity of π0 and π1 (and hence of π); see Lemma 2.1. For (3), consider x∈X0 and y∈X2∗. For every k∈K define F(k)=π0(k−1g2)x∈X1 andH(k)=π1(g1k)∗y∈X1∗. Fork1, k2∈K we have
hH(k1), F(k2)i=hy, π(g1k1k2−1g2)xi.
We view the continuous function F as an element of L2(K;X1). Its norm is less than supk∈Kkπ0(k−1g2)xk6Les`(g2)kxk. Similarly, we view H in the topological dual L2(Ω;X1)∗, and it has norm6Les`(g1)kykX2∗. We can compute
hH,(λ(µ)⊗idX1)(F)i= Z Z
K
hH(k1), F(k−12 k1)idµ(k2)dk1=hy, π(δg1µδg2)xi.
One deduces
|hy, π(δg1(µ−µ0)δg2)xi|6kλ(µ−µ0)⊗idX1k kFkL2(K;X1)kHkL2(K;X1)∗, which is less than
kλ(µ−µ0)kB(L2(K))L2es`(g1)es`(g2)kxk kyk,
because X1 is a Hilbert space. The lemma follows by taking the supremum over all x andy in the unit balls ofX1 andX2∗, respectively.
Remark 3.2. If we are in the setting of property (∗E) (that is, if X1 is a Banach space inE), then Lemma 3.1and its proof still holds, with (3) replaced by
d(δg1µδg2, δg1µ0δg2)6L2es`(g1)+s`(g2)kλ(µ−µ0)kB(L2(K;X1)).
We shall prove Theorem 1.3 for SL3(R) in the generality given by the previous lemma. So let dbe a distance on the compactly supported probability measures on G satisfying the three conditions (1)–(3) in the previous lemma.
We say that a probability measure ν on a compact group K is admissible if it is absolutely continuous with respect to the Haar measure onKand if the Radon–Nikodym derivative is strictly positive and is a coefficient of a finite-dimensional representation ofK. We say thatν is central if it belongs to the center of the convolution algebra of Borel measures onK.
Proposition 3.3. Denote by λK the Haar probability measure on K, seen as a probability measure on G. There existsC >0 such that,if s<14 and t:= 12−2s
>0,then d(λKδgλK, λKδg0λK)6 C
1−4sL2max(e−t`(g), e−t`(g0)). (3.1) For every admissible and central probability measureν onK,there is C(ν)∈Rsuch that, for every g∈G,
d(νδgλK, λKδgλK)6C(ν)L2e−t`(g). (3.2) This proposition easily implies the theorem. Indeed, the first half implies that there isP in the completion of (Pc(G), d) (which is contained inB(X0, X2) in our case) such that
d(λKδgλK, P)6 C
1−4sL2e−t`(g). More generally ifm0∈Pc(G), applying the same to
d0(m, m0) =d(m0m, m0m0)
(which satisfies the same assumptions thandwithL2replaced byL2esRifm0is supported in{g:`(g)=R}), we obtainm0P in the completion of (Pc(G), d) such that
d(m0λKδgλK,m0P)6 C
1−4sL2esR−t`(g). (3.3) Lemma 3.4. The map m07!m0P is lower-semicontinuous.
Proof. Let mi (resp. m0i), i∈I, be a net converging weak-∗ to m0 (resp. m00) and supported in a common compact subset ofG, say {g:`(g)6R}. For every g∈G, (3.3) yields
d(m0P,m00P)62 C
1−4sL2esR−t`(g)+d(m0λKδgλK, m00λKδgλK).
By the lower-semicontinuity ofd, we deduce d(m0P,m0
0P)62 C
1−4sL2esR−t`(g)+lim inf
i d(miλKδgλK, m0iλKδgλK),
which (by (3.3)) is bounded above by 4 C
1−4sL2esR−t`(g)+lim inf
i d(miP,m0
iP).
The lemma follows by making`(g)!∞.
The second half of the proposition implies that d(νδgλK, P)6
C
1−4s+C(ν)
L2e−t`(g),
ifν is an admissible and central probability measure onK. Using the convexity (1) and the lower-semicontinuity (2) ofd, we get that, for g1∈G,
d(νδg1λKδgλK, P)6 C
1−4s+C(ν)
L2e−t`(g)+t`(g1).
Making `(g)!∞, we obtain νδg1P=P. By the Peter–Weyl theorem, we can find a sequence νn of admissible and central probability measures on K converging weak-∗
toδ1. By Lemma3.4, we deduce that d(P,δg1P)6lim inf
i d(νiP,νiδg1P) = 0.
To summarize, ifmg is theK-biinvariant probability measure onKgK, we have proven that
d(mg, P)6 C
1−4sL2e−t`(g)
and limgd(δg1mg, P)=0 for everyg1∈G. If we consider the distance (m, m0)7!d(m, m0) form the image ofmby the inverse map,(3)we also have limgd(mgδg2, P)=0 for every g2∈G, and hence limgd(δg1mgδg2, P)=0. This proves the theorem.
It remains to prove Proposition3.3. As in Lafforgue’s original proof [13] (see also the exposition in [21]), the proof is based on the harmonic analysis in the compact groupK.
We introduce the subgroupsU,Ue⊂K of block-diagonal matrices
U=
∗ 0 0 0 ∗ ∗ 0 ∗ ∗
∩K and Ue=
∗ ∗ 0
∗ ∗ 0 0 0 ∗
∩K.
Note thatU andUe are both isomorphic to O(2).
(3) This new distance satisfies the same hypotheses asd.
Forδ∈[0,1] we introduce the following matrixkδ∈K with entry (1,1) equal toδ:
kδ=
δ −√
1−δ2 0
√1−δ2 δ 0
0 0 1
.
The fundamental inequality proven by Lafforgue in [13, Lemme 2.2] is that
Z Z
U×U
λ(ukδu0)−λ(uk0u0)du du0 B(L
2(K))
62|δ|1/2. (3.4) This implies more generally that if µ1 and µ2 are admissible probability measures onU, then
Z Z
U×U
λ(ukδu0)−λ(uk0u0)dµ1(u)dµ2(u0) B(L
2(K))
6C(µ1, µ2)|δ|1/2. (3.5) See [13] or [11, Proposition 2.1].
Forα, β, γ∈Rwith α+β+γ=0, we denote
D(α, β, γ) =
eα 0 0 0 eβ 0 0 0 eγ
.
Forα>0, we simply writeDα forD(2α,−α,−α). It has norme2αand`(Dα)=2α.
We start with the proof of (3.1). Denote by Λ the Weyl chamber, that is Λ ={(a1, a2, a3)∈R3:a1>a2>a3anda1+a2+a3= 0}.
For (a1, a2, a3)∈Λ denote
c(a1, a2, a3) =λKδD(a1,a2,a3)λK. By theKAK-decomposition, (3.1) is equivalent to the inequality
d(c(a1, a2, a3), c(a01, a02, a03))6 C
1−4sL2max(e−tmax(a1,−a3), e−tmax(a01,−a03)).
SinceDαcommutes with every element ofU, we can write λKδDαkδDαλK=λKδDαλUδkδλUδDαλK.
It therefore follows from (3.4) and the properties ofdin Lemma3.1that d(λKδDαkδDαλK, λKδDαk0DαλK)62L2e4sα|δ|1/2.
To make this formula more readable, we compute the KAK decomposition ofDαkδDα. Forδ=0, we have
Dαk0Dα=D(α, α,−2α)k0. (3.6) Forδ6=0, we have the lemma.
Lemma 3.5. For every r∈[α,4α] there are δ∈[0,1] and ur,α, u0r,α∈Ue such that δ6er−4α61 and
DαkδDα=ur,αD(r,2α−r,−2α)u0r,α.
Proof. For δ6=0, we have that g=DαkδDα is block diagonal with one eigenvalue e−2αand another block of the formDkDforD=diag(e2α, e−α) andkbeing an isometry.
In particular, kg−1k=e2α. If we define rα(δ)∈[0,∞) by kgk=erα(δ), we therefore have that
g∈U D(re α(δ),2α−rα(δ),−2α)U .e
By saying that the norm ofg is larger that the absolute value of its (1,1) entry, we get the desired inequality δe4α6erα(δ). It remains to show thatrα is surjective. But rα is continuous on the interval [0,1], so its image contains the interval [rα(0), rα(1)]=[α,4α].
We do not need it, but it is not hard to check thatrαis actually bijective from [0,1] onto [α,4α].
In particular, for every (a1, a2, a3),(a01, a02, a03)∈Λ satisfyinga3=a03, by applying the preceding lemma with−2α=a3=a03 we have that
d(c(a1, a2, a3), c(a01, a02, a03))|64L2(ea1/2+(1−2s)a3+ea01/2+(1−2s)a03). (3.7) Notice that, ifa2>−1, we have
1
2a1+(1−2s)a3=12(a1+a2+a3)+ 12−2s
a3−12a2612+ 12−2s a3. Therefore, (3.7) implies that
d(c(a1, a2, a3), c(a01, a02, a03))614L2e(1/2−2s)a3 ifa3=a03anda2, a02>−1. (3.8) If we apply the same for the distance d0(m, m0)=d(%∗m, %∗m0), where% is the Cartan automorphism
g7−!
0 0 1 0 1 0 1 0 0
(g−1)t
0 0 1 0 1 0 1 0 0
−1
,
we get that
d(c(a1, a2, a3), c(a01, a02, a03))614L2e−(1/2−2s)a1 ifa1=a01 anda2, a0261. (3.9) In particular, ifcr=c(r,0,−r) and 16r16r26r1+1,
d(cr2, cr1)6d(cr2, c(r2, r1−r2,−r1))+d(c(r2, r1−r2,−r1), cr1) 614L2(e−(1/2−2s)r2+e−(1/2−2s)r2).
s=t r=s
s=−1 s=0
Figure 1. The zig-zag path in the Weyl chamber Λ.
This implies, since
X
k>0
e−(1/2−2s)k6 3 1−4s, that, for everyr, r0>1,
d(cr, cr0)6 42
1−4sL2max(e−(1/2−2s)r, e−(1/2−2s)r0).
It follows easily from the above estimates that d(c(a1, a2, a3), c(a01, a02, a03))6 70
1−4sL2max(e−(1/2−2s) max(a1,−a3), e−(1/2−2s) max(a1,−a3)), which is exactly (3.1). The previous computations are best understood on a picture (see Figure 1): (3.8) expresses thatc is almost constant on lines of slope −12 in the region s>−1, whereas (3.9) expresses that c is almost constant on vertical lines in the region s60. These estimates are combined by the zig-zag path in Figure1.
We now move to the proof of (3.2). We start by a general lemma, valid for any pair of compact groupsU⊂K.
Lemma3.6. Every admissible probability measure ν on Kcan be written as ν1µfor admissible probability measures ν1 on K and µ on U.
Proof. By assumption, the Radon–Nikodym derivative dν/dk of ν is positive and is a coefficient of a finite-dimensional representationV ofK. Denote by CV the finite- dimensional space of real-valued matrix coefficients ofV, equipped (say) with theL∞(K)- norm. Letµn be a sequence of admissible probability measures onU converging weak-∗
to δe. Then Tn:CV3f7!f∗µn∈CV converges pointwise to the identity. Since CV has finite dimension, fornlarge enough this linear map is invertible and there is a sequence fn∈CV converging todν/dksuch thatTnfn=dν/dk. Sincedν/dkis positive, so isfn for nlarge enough. In other words, ν1=fndkis a probability measure such that ν1µn=µ, as requested.
Let us fix an admissible and central probability measure ν on K. Let ν=ν1µ be a decomposition given by the previous lemma. Since Dα commutes with every element ofU, one can write, forα>0 andδ∈[−1,1],
νδDαkδDαλK=ν1δDαµδkδλUδDαλK. By the convexity and lower-semicontinuity ofd, the distance
d(νδDαkδDαλK, νδDαk0DαλK) is therefore bounded by
sup
k,k0∈K
d(δkDαµδkδλUδDαk0, δkDαµδk0λUδDαk0).
By combining this inequality with the last point in Lemma3.1, (3.5), (3.6) and Lemma3.5, we get a constantC(ν) such that, for every αandr∈[α,4α],
d(νδur,αD(r,2α−r,−2α)λK, νδD(α,α,−2α)λK)6C(ν)L2er/2−(2−4s)α.
In particular and as for (3.8), ifa=(a1, a2, a3)∈Λ satisfiesa2>0 anda3=−2α, then there isua∈Ue such that
d(νδuaD(a1,a2,a3)λK, νδD(α,α,−2α)λK)6C(ν)L2e(1−4s)α. Let us apply the preceding to the distanced0(m, m0)=d(δuu−1
a m, δuu−1
a m0) for someu∈Ue, which satisfies the same assumptions asd. Note that, sinceνis central andD(α, α,−2α) commutes withuu−1a , we have
δuu−1
a νδuaD(a1,a2,a3)λK=δuνδD(a1,a2,a3)λK and
δuu−1
a νδD(α,α,−2α)λK=νδD(α,α,−2α)λK. Therefore, we obtain
d(δuνδD(a1,a2,a3)λK, νδD(α,α,−2α)λK)6C(ν)L2e(1−4s)α. (3.10)
